Poset Maps with Small Fibers

Theorem 11.4. ( Acyclic matchings via poset maps with small fibers)... 

For any poset map with small fibers P Q, the partial matching M[ip) is acyclic. Conversely, any acyclic matching on P can be represented as M (p) for some poset map with small fibers (p. 

On the other hand, by Theorem 11.2, for any acyclic matching on P there exists a linear extension L of P such that the elements a and u a) follow consecutively in L. Gluing a with u a) in this order yields a poset map with small fibers from P to a chain. ... 

In the proof of Theorem 11.4 we have actually constructed a poset map with small fibers into a chain. These maps are especially important, and we give them a separate name. 

We conclude our discussion of poset maps with small fibers by mentioning that this point of view yields a rich class of generalizations. Indeed, any choice of the set of allowed fibers will yield a combinatorial theory that could be interesting to study. One could, for instance, allow any Boolean algebra as a fiber. This would correspond to the theory of all coUapses, not just the elementary ones, which we get when considering the small fibers. One can take any other infinite family of posets. One prominent family is that of partition lattices IIn =i- What happens if we consider all poset maps with partition lattices as fibers ... 

Our main innovation in Section 11.1 is the equivalent reformulation of acyclic matchings in terms of poset maps with small fibers, as well as the introduction of the universal object connected to each acyclic matching. The patchwork theorem 11.10 is a standard tool, used previously by several authors. We think that the terminology of poset fibrations together with the decomposition theorem 11.9 give the patchworking particular clarity. 

Beyond the encoding of all allowed collapsing orders as the set of linear extensions of the universal object U (P, M), viewing the posets with small fibers as the central notion of the combinatorial part of discrete Morse theory is also invaluable for the structural explanation of a standard way to construct acyclic matchings as unions of acyclic matchings on fibers of a poset map. 

Proof. The role of the base space here is played by the poset Q, and the fiber maps gq are given by the acyclic matchings on the subposets ip q). The decomposition theorem tells us that there exists a poset map from P to the total space of the corresponding poset fibration, and that the fibers of this map are the same as the fibers of the fiber maps Qq. Since the latter are given by acyclic matchings, we conclude that we have a poset map from P with small fibers that corresponds precisely to the patching of acyclic matchings on the subposets 95 (g), for q Q. ... 

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